Adaptive solution of truss layout optimization problems with global stability constraints

Truss layout optimization problems with global stability constraints are nonlinear and nonconvex and hence very challenging to solve, particularly when problems become large. In this paper, a relaxation of the nonlinear problem is modelled as a (linear) semidefinite programming problem for which we describe an efficient primal-dual interior point method capable of solving problems of a scale that would be prohibitively expensive to solve using standard methods. The proposed method exploits the sparse structure and low-rank property of the stiffness matrices involved, greatly reducing the computational effort required to process the associated linear systems. Moreover, an adaptive ‘member adding’ technique is employed which involves solving a sequence of much smaller problems, with the process ultimately converging on the solution for the original problem. Finally, a warm-start strategy is used when successive problems display sufficient similarity, leading to fewer interior point iterations being required. We perform several numerical experiments to show the efficiency of the method and discuss the status of the solutions obtained

Layout optimization of pin-jointed truss structures with minimum frequency constraints

Controlling the frequency response of an engineering component or structure is important in the aerospace and automotive sectors and is a key consideration when seeking a new, more efficient, design for a given component. In this contribution, the standard truss layout optimization procedure is modified to incorporate semidefinite constraints to limit the minimum value of the first natural frequency. Since this increases the computational expense, and reduces the scale of problem that can be solved, a bespoke algorithm incorporating an adaptive ‘member adding’ procedure is proposed and applied to a number of benchmark example problems. It is demonstrated that this allows problems to be solved with relatively fine numerical discretisation, allowing modified structures with an acceptable minimum first natural frequency response to be successfully identified